7. If x and y are two vectors in Rn, n 2; 3 and z1y

7. If x and y are two vectors in Rn, n ¼ 2; 3 and z1y  x:

(a) Demonstrate kzk2 2 ¼ kxk2 2 þ kyk2 2  2x  y. (Hint: Use (3.5) and properties of inner products.)

(b) Show that if y < p denotes the angle between x and y, then cos y ¼
x  y kxk2kyk2 : ð3:51Þ
(Hint: the law of cosines from trigonometry states that c2 ¼ a2 þ b2  2ab cos y; ð3:52Þ
where

a, b, c are the sides of a triangle, and y is the radian measure of the angle between sides a and

b. Now create a triangle with sides x, y, and z.)

(c) Show that if y < p denotes the angle between x and y, then x  y ¼ 0 i¤ y ¼ 90, so x and y are ‘‘perpendicular.’’ (Note: The usual terminology is that x and y are orthogonal, and this is often denoted x ? y.)
Remark 3.48 Note that for n > 3, the formula in (3.51) is taken as the definition of the cosine of the angle between x and y, and logically represents the true angle between these vectors in the two-dimensional plane in Rn that contain them. As was noted in the section on inner products, the derivations in

(a) and

(b) remain true for a general inner product and associated norm, and hence the notion of ‘‘orthogonality’’ can be defined in this general context.